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Sekulić D. et al. The Effect of Stiffness and Damping of the Suspension System Elements on the Optimisation of the Vibrational Behaviour of a Bus

THE EFFECT OF STIFFNESS AND DAMPING OF THE SUSPENSION SYSTEM ELEMENTS ON THE OPTIMISATION OF THE VIBRATIONAL BEHAVIOUR OF A BUS

Dragan Sekulić1, Vlastimir Dedović2 1,2 University of Belgrade, Faculty of Transport and Traffic Engineering, Vojvode Stepe 305, 11000 Belgrade, Serbia

Received 16 May 2011; accepted 1 September 2011

Abstract: The effects of spring stiffness and shock absorber damping on the vertical acceleration of the driver’s body, suspension deformation and dynamic wheel load were investigated, with the purpose to define recommendations for selecting oscillation parameters while designing the suspension system of a (intercity) bus. Oscillatory responses were analysed by means of a bus oscillatory model with linear characteristics and three degrees of freedom, with excitation by the Power Spectral Density (PSD) of the roughness of asphalt-concrete pavement in good condition. The analysis was conducted through a simulation, in frequency domain, using statistical dynamics equations. A programme created in the software pack MATLAB was used to analyse the transfer functions, spectral density and RMS of oscillatory parameters. The results of the analysis show that the parameters which ensured good oscillatory comfort of the driver were conflicting with the parameters which ensured the greatest stability of the bus and the corresponding wheel travel. In terms of the driver’s oscillatory comfort, the bus suspension system should have a spring of small stiffness and a shock absorber with a low damping coefficient. In terms of active safety, it should have a spring of small stiffness and a shock absorber with a high damping coefficient, while minimum wheel motion requests for springs of great stiffness and shock absorbers with a high damping coefficient.

Keywords: bus, oscillatory behaviour, spring, shock absorber, simulation.

1 Corresponding author: d.sekulic@sf.bg.ac.rs

1. Introduction

The suspension system is one of the main vehicle systems, significant for the realisation of comfort, stability and safety parameters. The main purpose of this system is to increase the comfort of vehicle occupants (passengers and drivers), to maintain the contact between the tyre and the road surface and to eliminate (minimize) dynamic forces which act on the load bearing vehicle structure and road surface along which the vehicle is moving.

Elastic and damping elements are built into the suspension system for the above- mentioned purposes can be successfully achieved. Pneumatic spring elements (also known as bellows or pneumatic balloons) are present as elastic elements in the suspension system of modern buses, while hydraulic shock absorbers are damping elements.

Some of the main advantages of pneumatic system compared to other suspension systems (featuring elastic springs, combined

UDC: 629.341.027.421

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suspension system) are the preservation of the same floor height of buses even with different static and dynamic loads, the same oscillation frequency of sprung mass irrespective of the bus load and greater oscillatory comfort. The disadvantages of the system are its relative complexity and higher price.

The choice of the suspension system parameters comes down to the resolution of different, conflicting demands. When developing the suspension system, an optimal compromise has to be made between comfort, rattle space and the variation of forces acting in contact between tyre and road surface.

The aim of this paper is to determine the spring stiffness and shock absorber damping values of the bus suspension system, needed to have acceptable oscillatory behaviour. Three important oscillatory parameters (dr iver ’s ver t ica l accelerat ion, bu s suspension deformation and dynamic wheel load) in frequency domain were analysed. This type of analysis allows the choice of the values of oscillatory parameters of the bus suspension system depending on different excitation frequency values. Similarly, the analysis facilitates the choice of the oscillatory parameter values for the excitation frequency range which exerts considerable influence on the oscillatory behaviour of the bus, which is, in turn, of great importance when designing the bus suspension system.

2. Oscillatory Model of the Bus

A linear oscillatory model of the bus used in simulation, with three degrees of freedom, is shown in Fig. 1. Table 1 provides the meaning of the values from the same figure, together with parameters used in the simulation.

Fig. 1. Linear Oscillatory Model of the Bus

Table 2 shows calculated eigenvalues (characteristic values), undamped natural frequencies and damped natural frequencies of the oscillatory system for the parameters in Table 1. The computed values of natural frequencies of oscillation are close to the values of typical oscillation frequencies of the bus in service (Dedović, 2004; Demić and Diligenski, 2003).

The differential equations of motion for the oscillatory system in matrix form are formulated as follows, Eq. (1):

(1)

where:

International Journal for Traffic and Transport Engineering, 2011, 1(4): 231 – 244

233

Table 1 Bus Parameters

Bus parameter Value

- mass of the driver and the seat 100 (kg)

- driver’s seat suspension spring stiffness 25000 (N/m)

- driver’s seat shock absorber damping 1000 (Ns/m)

- sprung bus mass on front axle 4500 (kg)

- front axle bellows stiffness 300000 (N/m)

- front axle shock absorbers damping 20000 (Ns/m)

- unsprung front axle mass 500 (kg)

- bus tyre stiffness 1600000 (N/m)

- bus tyre damping 150 (Ns/m)

- bus excitation

Table 2 Characteristic Values, Undamped Natural Frequencies and Damped Natural Frequencies of the Oscillatory System

Concentrated mass Eigenvalues Undamped natural frequencies (Hz) Damped natural frequencies (Hz)

Driver and seat -5.1634 ±15.1915i 2.5536 2.4178

Bus body -1.5598 ± 7.3436i 1.1949 1.1688

Axle -20.7600 ±56.9619i 9.6491 9.0658

Sekulić D. et al. The Effect of Stiffness and Damping of the Suspension System Elements on the Optimisation of the Vibrational Behaviour of a Bus

234

- symmetric inertia matrix, damping matrix and stiffness matrix

- stiffness and damping matrices of the input (excitation)

- the column vector of generalised coordinates, general i sed velocities and generalised accelerations

- the column vector of input vertical motion and velocity (excitation).

The column vector of the oscillatory system generalised coordinates is equal to (Eq. (2)):

(2)

The transfer functions of the oscillatory system motion, in complex form, are formulated as follows, Eq. (3):

(3)

where:

f – is the temporal frequency of excitation (Hz)

i – is the imaginary unit.

The transfer functions of the driver’s vertical acceleration, bus suspension deflection and dynamic wheel load in complex form are represented by following expressions (Eqs. (4), (5) and (6)):

(4)

(5)

(6)

3. Excitation from Road Surface

The PSD of the roughness of asphalt- concrete pavement in good condition, as a function of temporal excitation frequency, based on (Wong, 2001), is expressed as (Eq. (7)):

(7)

where:

f – is the temporal frequency of excitation (Hz)

V – is the speed of the bus (m/s)

Csp – is the roughness coefficient (m)

N – is the wave number (-).

Table 3 provides the values of the roughness coefficient Csp and wave number N for the asphalt- concrete pavement in good condition, according to (Wong, 2001).

Table 3 Roughness Coefficient and Wave Number for the Asphalt-Concrete Pavement

Pavement type and condition N Csp

Asphalt-concrete (good) 2.1 4.8*10

-7

In the frequency range 0.5 Hz to 50 Hz (Kawamura and Kaku, 1985) the pavement roughness has the most significant impact on the vehicle’s oscillatory behaviour. The PSD of the asphalt-concrete pavement roughness for the parameters in Table 3 and within the above-referenced frequency range is shown in Fig. 2.

International Journal for Traffic and Transport Engineering, 2011, 1(4): 231 – 244

235

4. Results of the Simulation

The responses of the linear oscillatory bus model are obtained using the Eq. (8):

(8)

where:

- is the PSD of the considered

response of the oscillatory model (oscillatory parameter)

- the root square of the transfer

function modulus of the considered response of the oscillatory model

- the PSD of pavement roughness.

The paper analyses the effects of spring

stiffness and shock absorber damping of the bus suspension system on three oscillatory parameters – the driver’s vertical acceleration, suspension deformation and dynamic wheel load, at a constant bus speed of 100 km/h.

Figs. 3a-3c show the transfer functions and Figs. 4a-4c show the PSD of the considered oscillatory parameters for constant shock absorber damping (b2=20000 Ns/m) and different values of spring stiffness (c2) of the bus suspension system.

The amplitudes of the driver’s vertical acceleration rose with an increase of s